Cacti and filtered distributive laws

CACTI AND FILTERED DISTRIBUTIVE LAWS

arXiv:1109.5345v1 [math.AT] 25 Sep 2011

VLADIMIR DOTSENKO AND JAMES GRIFFIN

Abstract. Motivated by the second author’s construction of a classifying space for the group of pure symmetric automorphisms of a free product, we introduce and study a family of topological operads, the operads of based cacti, defined for every pointed topological space (Y, •). These operads also admit linear versions, which are defined for every augmented graded cocommutative coalgebra C. We show that the homology of the topological operad of based Y cacti is the linear operad of based H∗ (Y )-cacti. In addition, we show that for every coalgebra C the operad of based C-cacti is Koszul. To prove the latter result, we use the criterion of Koszulness for operads due to the first author, utilising the notion of a filtered distributive law between two quadratic operads. We also present a new proof of that criterion which works over a ground field of arbitrary characteristic.

1. Introduction One of the most famous algebraic operads of topological origin is the operad of Gerstenhaber algebras, which is the homology operad of the topological operad of little 2-disks [9, 16]. The k th component of the operad of little 2-discs is homotopy equivalent to the configuration space of k ordered points in R2 whose fundamental group is the pure braid group on k strands. One natural way to generalise braid groups is to consider configurations of subsets that have more interesting topology than points. The simplest example of these “higher-dimensional” versions of braid groups is given by “groups of loops”, the nth one being the group of motions of n unknotted unlinked circles in R3 bringing each circle to its original position. Alternatively, these groups can be viewed as groups of pure symmetric automorphisms of the free group with n generators, that is automorphisms sending each generator to an element of its conjugacy class. The integral cohomology of these groups was computed by Jensen, McCammond and Meier in [19]; that paper also contains references and historical information on this group. The description of the cohomology algebras in [19] looks very similar to that for pure braid groups [2]. Moreover, as a symmetric collection, the collection of cohomology algebras is isomorphic to Com ◦ PreLie1 which bears striking resemblance with the isomorphism e2 ' Com ◦ Lie1 for the operad of Gerstenhaber algebras. However, there is no natural operad structure on the collection of homology groups of the groups of loops. In [18], the second author computed the cohomology of the groups of pure symmetric automorphisms in a different way, as a particular case of a much more general result: for an arbitrary n-tuple of groups (G1 , . . . , Gn ), he computed the cohomology of the Fouxe-Rabinovitch group FR(G) of partial conjugation automorphisms of the free product G = G1 ∗ · · · ∗ Gn . For that, he used a construction of a classifying space of that group via a moduli space of “cactus products” of the classifying 1


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spaces Yi = BGi . In the case when G1 = G2 = . . . = Gn , these spaces form a symmetric collection, but alas do not form a topological operad either. However, it turns out that they admit a slight modification that carries a structure of a topological operad; the required change is that one of the spaces Yi is chosen as the base and is required to sit at the root of each cactus. We call the modified space the space of based Y -cacti. The goal of this paper is to understand the algebra and topology of this operad. For homology with coefficients in a field, we show that the homology operad of the operad of based Y -cacti is obtained from the homology coalgebra of Y by a formal algebraic procedure that only uses the augmentation and the coproduct; thus, it is defined for every graded cocommutative coalgebra C, not necessarily the homology coalgebra of a topological space. Remarkably, for every coalgebra C this defined operad is Koszul. To prove that, we use filtered distributive laws between operads, as defined by the second author in [10]. One immediate consequence of our results is that, for Y = S 1 , the homology operad of based Y -cacti is isomorphic, as an Smodule, to Perm ◦ PreLie1 , which, given that the operad of associative permutative algebras Perm encodes commutative algebras with additional structure, may be naturally thought of as an “operad-compatible improvement” of the result of [19] mentioned above. Our constructions are defined over a field of arbitrary characteristic, and our results on operads of based cacti hold in that generality. However, even the distributive law criterion for Koszulness, let alone its filtered generalisation, has only been available in zero characteristic, since the known proofs [10, 32] rely on the K¨ unneth formula for symmetric collections. Using the shuffle operads technique [12, 13], we were able to obtain a characteristic-independent proof of this criterion. The paper is organised as follows. In Section 2, we recall necessary background information that we use throughout the paper. In Section 3, we define the topological operads of based cacti and discuss its connections both with automorphism groups of free products and with other known topological operads. The homology operad for the operad of based cacti is computed in Section 4. In Section 5, we discuss filtered distributive laws between quadratic operads. Section 6 shows how to use filtered distributive laws to prove the Koszul property for the linear operads of based cacti, and also discuss its applications to the operad of post-Lie algebras and the operad of commutative tridendriform algebras. 2. Trees, coalgebras, operads All “linear” objects in this paper (algebras, coalgebras, operads) will be enriched in a certain symmetric monoidal category (C, ⊗, σ, I), usually the category Vect of vector spaces or the category gVect of graded vector spaces (over some field k; unless otherwise specified, we do not make any assumptions on its characteristic). Whenever appropriate, we assume vector spaces to be finite-dimensional, or possessing an additional N-grading with finite-dimensional homogeneous components; this allows to approach tensor constructions and duals with ease, freely pass between an algebra and its dual coalgebra etc. 2.1. Y-labelled trees. A tree is an acyclic connected graph and a rooted tree is a tree with a chosen vertex, the root. A rooted tree may be directed: every edge → in such a way that the minimal path from w to {v, w} may be oriented to − vw the chosen vertex contains {v, w}. By the acyclicity of the tree this must hold for


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→ and − → The edges may be seen to be directed ‘away exactly one of the choices − vw wv. from the roots’. We denote by E(T ) the set of edges of a tree T . Suppose that T is a tree with vertex set V . Let Y = (Yi )i∈V be a V -tuple of topological spaces. Then a Y-tree is a rooted tree with an edge labelling where the → − edge ij is labelled by an element of Yi . For a space Y as shorthand we define a Y -tree to be a Y-tree where the V -tuple Y is constantly Y . Then the edge labelling is a map from the edge set E to the space Y . Meanwhile a Y -forest is a Y-tree where Y is the V -tuple with Y0 ∼ = {•}, where 0 is the root vertex and Yv ∼ = Y for any other vertex. The naming makes sense because by removing the root 0 and all adjacent vertices we are left with a disjoint union of Y -trees; the root of each tree is the unique vertex adjacent to 0 and the edge labelling is inherited. To a rooted tree T we define the level l(T ) to be the number of non-trivial directed paths in T . So for a corolla with root 1 and k − 1 other vertices the level −−−−→ is k − 1, for a tree with root 1 and edges i(i + 1) for i = 1, . . . , k − 1 the level is k(k − 1)/2. The level allows one to filter the set of Y-trees. 2.2. Coalgebras. A coalgebra is an object C of C equipped with a comultiplication ∆ : C → C ⊗ C and a counit : C → I satisfying the conventional coassociativity and counit P axioms. For the comultiplication, we often use Sweedler’s notation ∆(c) = c(1) ⊗ c(2) . An augmented coalgebra is a coalgebra C equipped with a coalgebra homomorphism γ : I → C such that γ = 1. A cocommutative coalgebra is a coalgebra satisfying σ∆ = ∆. Our main focus will be on graded augmented cocommutative coalgebras, that is augmented cocommutative coalgebras in gVect. The main source of such coalgebras relevant for our purposes is topology: the homology coalgebra of a pointed topological space (Y, •) is a graded augmented cocommutative coalgebra. An augmented coalgebra in Vect or gVect naturally splits into a direct sum of vector spaces C = k 1 ⊕C, where 1 = γ(1), C = ker(). 2.3. Operads. For details on operads we refer the reader to the book [25], for details on Gr¨ obner bases for operads — to the paper [13]. In this section we only recall the key notions used throughout the paper. By an operad (enriched in a symmetric monoidal category (C, ⊗, σ, I)) we mean a monoid in one of the two monoidal categories: the category of symmetric C-collections equipped with the composition product or the category of nonsymmetric C-collections equipped with the shuffle composition product. The former kind of monoids is referred to as symmetric operads, the latter — as shuffle operads. We always assume that our collections are reduced, that is, have no elements of arity 0. A good rule of thumb is that all operads defined in this paper are symmetric operads, but for computational purposes it is useful to treat them as shuffle operads. This does not lose any information except for the symmetric group actions, since the forgetful functor O 7→ O f is monoidal and one-to-one on objects (and therefore for tasks that can be formulated without the symmetric group actions, e.g. computing bases and dimensions of components, proving the Koszul property etc., we can choose arbitrarily whether to work with a symmetric operad or with its shuffle version). In the “geometric” setting, C will usually be the category of sets, or topological spaces, or pointed topological spaces, in the “linear” setting — the category of vector spaces (in which case symmetric collections are usually called S-modules), or the category of graded vector spaces or chain complexes (in which case symmetric collections are called differential graded S-modules). A linear symmetric operad can also be


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thought as of collection of spaces of operations of some type, and therefore can be defined via its category of algebras, i.e. vector spaces where these operations act, via identities between operations acting on a vector space. In the linear setting, a very useful technical tool for dealing with (shuffle) operads is given by Gr¨ obner bases. More precisely, similarly to associative algebras, operads can be presented via generators and relations, that is as quotients of free operads F (V ), where V is the space of generators. The free shuffle operad generated by a given nonsymmetric collection admits a basis of “tree monomials” which can be defined combinatorially; a shuffle composition of tree monomials is again a tree monomial. In addition to the “arity” of elements of a free operad, there is the notion of weight, similar to grading for associative algebras: we define the weight of a tree monomial as the number of generators used in this tree monomial. Weight is well behaved under composition: when composing several tree monomials, the weight of the result is equal to the sum of their weights. For an arbitrary operad O = F (V )/(R) whose relations R are weight-homogeneous, the weight descends from the free operad F (V ) on O; the subcollection of O consisting of all elements of weight k is denoted by O(k) . There exist several ways to introduce a total ordering of tree monomials in such a way that the operadic compositions are compatible with that total ordering. There is also a combinatorial definition of divisibility of tree monomials that agrees with the naive operadic definition: one tree monomial occurs as a subtree in another one if and only if the latter can be obtained from the former by operadic compositions. A Gr¨ obner basis of an ideal I of the free operad is a system S of generators of I for which the leading monomial of every element of the ideal is divisible by one of the leading terms of elements of S. Such a system of generators allows to perform “long division” modulo I, computing for every element its canonical representative. There exists an algorithmic way to compute a Gr¨obner basis starting from any given system of generators (“Buchberger’s algorithm for shuffle operads”). A part of the operad theory which provides one of the most useful known tools to study homological and homotopical algebra for algebras over the given operad is the Koszul duality for operads [17]. Proving that a given operad is Koszul instantly provides a minimal resolution for this operad, gives a description of the homology theory and, in particular, the deformation theory for algebras over that operad etc. There are a few general methods to prove that an operad is Koszul; one of the simplest and widely applicable methods [13, 12] is to show that a given operad has a quadratic Gr¨ obner basis (as a shuffle operad); this provides a sufficient (but not necessary) condition for Koszulness of an operad. If an operad is Koszul, it necessarily is quadratic, that is has weight-homogeneous relations of weight 2. The operads that serve as “building blocks” for operads considered throughout the paper are mostly well known: Com (commutative associative algebras), Lie (Lie algebras), As (associative algebras), Leib (Leibniz algebras [26]), Zinb (Zinbiel algebras [27]), Perm ([associative] permutative algebras [8]), NAP (nonassociative permutative algebras [24], closely related to “right-commutative magma” [15]). All these operads are Koszul, and have a quadratic Gr¨obner basis. 2.4. Polynomial functors. As we said before, some of our constructions exist both in a “geometric” and a “linear” setting, and are related to each other via the homology functor (which assigns to a topological space Y the graded cocommutative coalgebra H∗ (Y )). To make additional structures transfer easily, we use basic


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concepts of the theory of polynomial functors. A polynomial functor is a notion that categorifies the notion of a polynomial, and more generally of a formal power series. Polynomial functors provide a useful uniform language to deal with categorical constructions that have “a polynomial flavour”, e.g. when computing sums and products in appropriate categories over specified sets indexing summands/factors in a way that keeps track of the intrinsic structure of the indexing sets. In precise words, a diagram of sets and set maps s

p

t

I ←− E −→ B −→ J

(2.4.1)

gives rise to a polynomial functor F : Set /I → Set /J defined by the formula (2.4.2)

s∗

p∗

t

! Set /J. Set /I −→ Set /E −→ Set /B −→

Here ∗ and ! denote, respectively, the right adjoint and the left adjoint of the pullback functor ∗ . More explicitly, the functor is given by X Y (2.4.3) [f : X → I] 7−→ f −1 (s(e)), b∈B e∈p−1 (b)

where the last set is considered to be over J via t! . Here one can replace Set by another category where all the appropriate notions make sense. For our purposes, it is enough to consider the case I = J = ∗, in which case the corresponding functors were referred to as polynomial functors in [31], and are called polynomial functors in one variable in more recent literature. For a systematic introduction to polynomial functors, we refer the reader to the paper [23] and the notes [22] that reflect the state-of-art of the theory. 3. The operad of cacti 3.1. The operad NAPY . Let Y be a set and let NAPY (n) be the set of Y -trees with vertex set [n] = {1, . . . , n}. When Y is a singleton set this is just the set of rooted trees which we denote RT(n). The symmetric group Sn acts on NAPY (n) by permuting elements of the vertex set. For a given rooted tree the set of Y labellings is equal to Hom(E, Y ) = Y E . Since the number of edges of a tree on {n} is always n − 1, the set of Y -labellings is in turn isomorphic to Y n−1 . Hence a (3.1.1) NAPY (n) ∼ Y n−1 . = T ∈RT(n)

In this way if Y is a topological space then we may also apply a topology to NAPY (n) using the product topology on Y n−1 . Now let T1 ∈ NAPY (n) and T2 ∈ NAPY (m) and i ∈ [n]. We may define a composition T1 ◦i T2 ∈ NAPY (n + m − 1) by first identifying the root of T2 with the vertex i in T1 . This is a tree and may be rooted by taking the root of T1 . The edge set is equal to the union E(T1 ) q E(T2 ) of the edge sets of T1 and T2 and so one inherits an edge labelling by elements of Y . It has the vertex set (3.1.2)

{1, . . . , i − 1} q {1, . . . , m} q {i + 1, . . . , n}.

We then relabel the vertices by elements of [n + m − 1] using the isomorphism which fixes {1, . . . , i − 1}, shifts the set {1, . . . , m} to {i, . . . , m + i − 1} and shifts {i + 1, . . . , n} to {m + i, . . . , m + n − 1}. This gives a rooted Y -tree on the vertex set {1, . . . , n + m − 1} and so an element T1 ◦i T2 ∈ NAPY (n + m − 1).


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Proposition 3.1. Let Y be a set, then the maps (3.1.3)

◦i : NAPY (n) × NAPY (m) → NAPY (n + m − 1)

for i = 1, . . . , n give the collection NAPY an operad structure. The operad is generated by its binary operations: 2O y

(3.1.4)

1O

and

1

z

2

for y, z ∈ Y and these satisfy the quadratic relation (3.1.5)

2O y 1

◦1

2O 1

z

=

2O

z

◦1

1

2O y

! .(23).

1

Proof. Let T1 , T2 and T3 be Y -trees in NAPY (n1 ), NAPY (n2 ) and NAPY (n3 ) respectively. Let i < j ∈ [n1 ] and k ∈ [n2 ]; we must show that the two associativity relations hold; (3.1.6)

(T1 ◦j T2 ) ◦i T3 = (T1 ◦i T3 ) ◦j+n3 −1 T2

and (3.1.7)

T1 ◦i (T2 ◦k T3 ) = (T1 ◦i T2 ) ◦k+i−1 T3 .

In both cases we are gluing together trees by identifying vertices — in the first we identify the roots of T2 and T3 with the vertices j and i of T1 respectively — whilst in the second the root of T2 is joined to vertex i of T1 and the root of T3 is identified with vertex k of T2 . The only complication is that when two trees are composed their vertices are renumbered: this change is taken into account in the right hand side of each equation. In both cases the edge set of the resulting tree is the union of the edge sets of the three component trees, hence the Y -labellings on both sides of each equation are equal. It remains to make the routine check that the vertex labels in each side of each equation agree, this is no more complicated than the analogous check in the associative operad. Now we show that the operad is generated by operations of arity 2. Let T ∈ → − → − NAPY (n) be any Y -tree and let ij be a leaf of T ; let y be the label of ij . By applying a permutation if necessary we may assume that i = n − 1 and j = n. −−−−−→ Letting T 0 be the Y -tree in NAPY (n − 1) given by removing the edge (n − 1)n and the vertex n, we have that (3.1.8)

T = T 0 ◦n−1

2O y

.

1 Therefore any Y -tree may be written as compositions of trees with two vertices and a permutation and so NAPY is generated in arity 2. The relation (3.1.5) is to seen to hold by evaluating each side of the equation to find the same Y -tree (3.1.9)

2 ^== 3 = @ . y z 1


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The above theorem gives quadratic relations in the binary generators, the Corollary 6.7 will show that these suffice to present the operad. Remark 3.1. The operads NAPY are functorial in sets Y , in fact NAP(−) (n) is a polynomial functor given by the diagram a (3.1.10) ∗← E(T ) → RT(n) → ∗. T ∈RT(n)

Both the operad maps and the proof above work on the level of the polynomial itself, hence for any appropriate category one may use the polynomial to give a family of operads NAP(−) . For instance this means that if Y is also equipped with a topology then NAPY is a topological operad. In Section 4 we will consider the operads NAPD where D is a graded vector space. Remark 3.2. When Y is a single point {•}, the operad NAPY is the usual operad NAP. Let us finish this section with a few words on NAPY -algebras. One convenient way to think of them is via the “right regular module”, since the defining relations say that all the right multiplications R(y, b) : a 7→

(3.1.11)

y

2O

(a, b)

1 commute with each other. Somewhat more precisely, let A be an object in a symmetric monoidal category C, and let (3.1.12)

f : Y × A → HomC (A, A)

be a map whose image is an abelian submonoid. Then A is a NAPY -algebra enriched in C with the structure maps given by (3.1.13)

y

2O

(a, b) = f (y, b).a.

1 This way to approach NAPY -algebras gives a source of examples based on Permalgebras with a family of maps as follows. Example 3.1. Let (A, ·) be a Perm-algebra encriched in a symmetric monoidal category C, and let gy , y ∈ Y be a family of maps in HomC (A, A) (note that these maps may be arbitrary, not necessarily algebra homomorphisms). Then A is a NAPY -algebra enriched in C with the structure maps given by (3.1.14)

y

2O

(a, b) = a · gy (b).

1 One more observation we want to mention in this section is that the construction of the free NAP-algebra mentioned in [24] admits an immediate generalisation to the case of NAPY -algebras: the free NAPY -algebra enriched in Set with the generating set V admits a realisation as the set of Y -trees whose vertices carry labels from V , with the product defined in the same way as we defined the composition in the operad: (3.1.15)

y

2O 1

(a, b) = (

y

2O 1

◦1 a) ◦2 b.


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In this composition the root of b is joined to the root of a by an edge labelled by y; the new root is taken to be the root of a. 3.2. The operad of based cacti. Let V be a set and Y be a V -tuple of pointed → − spaces. Let T be a Y-tree with root r ∈ V and suppose that ij is an edge of T → − where i 6= r. Suppose further that ij is labelled by the basepoint • ∈ Yi . Then we → − say that ij is a reducible edge and that T is reducible. Since i is not the root there → − is a unique incoming edge ki which is labelled by some y ∈ Yk . We define Tij to be − → → − the Y-tree given by removing the edge ij and adding the edge kj with the label y ∈ Yk . We say that Tij is a reduction of T . Definition 1. Let V be a finite set and Y be a V -tuple of pointed spaces. Then the space of based Y-cacti, BCactY is the topological space given by quotienting → − out by the relation T ∼ Tij for any T with a reducible edge ij . Now let V0 be the set V ∪ {0} and let Y0 be the V0 -tuple given by adjoining Y0 = {•} to the V -tuple Y. Then we define the space of Y-cacti, CactY to be the subspace of BCactY0 consisting of the trees with root 0. Remark 3.3. For each Y-cactus T ∈ BCactY one may define the space ` v∈V Yv (3.2.1) Y(T ) = , → − yij ∼ •j | ij ∈ E(T ) → − where yij ∈ Yi is the label of the edge ij and •j is the basepoint of Yj . Note that this realisation is invariant across equivalences T ∼ Tij . If each space Yi is path connected then this space is homotopy equivalent to the wedge product of the spaces Yv for v ∈ V . These spaces are called cactus products and were studied by the second author in [18]. There it was shown that the space CactY of such products has interesting homotopical properties, in particular if the spaces Yi are classifying spaces for groups Gi then CactY is a classifying space for the FouxeRabinovitch group FR(G) of partial conjugation automorphisms of the free product G = ∗i∈V Gi . An example of a cactus product:

(3.2.2)

Note that if v is the root of the tree T then the space Yv must always be at the ‘base’ of the diagram. The appearance of the diagram explains the term ‘based Y-cactus’. We also see the reason for adjoining a point space Y0 ; this removes the base space; the space Y0 acts as a basepoint. Remark 3.4. Recall that the level of a rooted tree is the number of non-trivial → − → − directed paths. When ki and ij are edges of a rooted tree T , the rooted tree T 0 − → → − given by removing ij and then adding kj has strictly lesser level. Indeed if P is the unique path joining vertices v and w in T 0 , then there is a unique path joining v and w in T . But the number of paths in T is strictly larger because there is a path joining i and j in T but not in T 0 . So for any Y-tree T one may use the reductions T ∼ Tij repeatly until there are no reducible edges remaining. Since


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the level reduces each time this process must terminate. It is easy to check that it → − does not matter what order the reductions T ∼ Tij are applied because if ab and → − cd are two reducible edges then (Tab )cd = (Tcd )ab . Hence for each Y -labelled tree there is a unique equivalent tree which can not be reduced any further. Therefore BCactY is isomorphic to the set of irreducible Y-trees. Definition 2. Let (Y, •) be a pointed space. For n ≥ 1, we define the space BCactY (n) to be the space of based cacti on the n-tuple Y = (Yi ∼ = Y )i=1,...,n . The action of Sn on {1, . . . , n} makes this into a symmetric collection. Theorem 3.2. Let (Y, •) be a pointed space. The equivalence relation ∼ generated by reductions T ∼ Tij is compatible with the operad maps of NAPY . Hence the quotient collection BCactY has an operad structure inherited from NAPY . Furthermore the equivalence relation ∼ is generated as an operad ideal by the single relation 3O (3.2.3)

2 ^== 3 = @ . y y 1

•

2O y

=

1 → − Proof. Let T ∈ NAPY (n) be a Y -tree with reducible edge ij ; let T 0 ∈ NAPY (m) be any other Y -tree. Then for any k ∈ [n] and l ∈ [m] the products T 0 ◦l T → − are both given by identifying vertices. The edge ij still exists in each product −→ although it may have been relabelled, to i0 j 0 say. The label in Y is still the point •. −→ Furthermore i0 is not the root in either product so i0 j 0 is a reducible edge giving the reductions (3.2.4)

(3.2.5)

T ◦k T 0

and

(T ◦k T 0 ) ∼ (T ◦k T 0 )i0 j 0

and

(T 0 ◦l T ) ∼ (T 0 ◦l T )i0 j 0 .

The reductions are also closed under the symmetric actions: for σ ∈ Sn the edge −−−−−→ (iσ)(jσ) is reducible in T σ. This shows the first part and in particular that BCactY is an operad. We will now show that all reductions T ∼ Tij are obtainable from the reduction (3.2.3) of 3O (3.2.6)

•

2O . y 1

→ − We must show that any reducible Y -tree T , with reducible edge ij say, is contained → − in the ideal in NAPY generated by (3.2.6). Let ki be the unique edge incoming to i. By applying a permutation we may assume that k = 1, i = 2 and j = 3. The essential idea of the proof is that since (3.2.6) is a subtree, the tree T may be → − written as a composition of (3.2.6) and other Y -trees. Removing the edges 12 and → − 23 from T leaves three connected components; T1 contains 1, T2 contains 2 and T3


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→ − → − contains 3. In effect we have partitioned the edge set of T into {12, 23}, E(T1 ), E(T2 ) and E(T3 ). Then we may express T as y • (3.2.7) T = T1 ◦1 ( 1 / 2 / 3 ) ◦3 T3 ◦2 T2 .σ, where σ is a permutation relabelling the vertices.

Remark 3.5. The Corollary 6.7 to Theorem 6.6 states that NAPY is binary quadratic. Along with the Theorem above this shows that BCactY is also binary quadratic. In the spirit of how we approached NAPY -algebras, a BCactY -algebra enriched in a symmetric monoidal category C is a NAPY -algebra enriched in C where the 2O operation • is associative, and 1 (3.2.8)

f (y,

2O

•

(a, b)) = f (y, a) ◦ f (y, b).

1 3.3. The fundamental groupoid of BCactY . Let Y be a topological space and let P be a subset of Y . We define the fundamental groupoid π1 (Y, P ) to be the groupoid with objects the points p ∈ P and morphisms the homotopy classes of paths in Y which start and end in elements of P . The composition is by concatenation of paths and the units are supplied by the constant paths. So if (Y, •) is a pointed space then π1 (Y, {•}) is the fundamental group of Y . Let (Y, •) be a pointed space and let P ∈ Y be a set of points which contains • and such that each path connected component of Y contains a single point of P . This may be seen as a section of the map (3.3.1)

(Y, •) → (π0 (Y ), π0 (•)).

Then by the functoriality of BCact(−) there is a pair of operad maps / BCact , (3.3.2) BCact o Y

P

which serves to pick out a single element in each path connected component of BCactY . The fundamental groupoid functor preserves products and colimits and so π1 (BCactY ; BCactP ) is an operad in the category of groupoids. From now on we will restrict Y to be a path connected space, so P = {•}. In this case BCactP ∼ = Perm, the operad for permutative algebras — each of the n elements is given by a corolla. So we see that BCactY (n) is made up of n components and the action of Sn gives isomorphisms between them. Denote by BCactY (n)r the component consisting of trees with root r. Proposition 3.3. The fundamental group of BCactY (n)1 is presented by generg ators αij for i = 1, . . . , n, j = 2, . . . , n with i 6= j and g ∈ π1 (Y, •), along with relations (3.3.3) (3.3.4)

g h gh αij αij = αij , g h αij , αik = e

for distinct i, j, k; (3.3.5)

g h αij , αkl =e


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for distinct i, j, k, l; and g g h αij αik , αjk = e

(3.3.6) for distinct i, j, k.

Proof. We defined the cactus operads BCactY by adding certain relations T ∼ Tij → − for trees T with a reducible edge ij . The relations come in families: for a fixed tree → − → − T with a fixed edge ij where i is not the root, a Y -tree is reducible if ij is labelled by the point • ∈ Y and the remaining n − 2 edges are labelled by any element in Y , so there is a family of relations parametrised by {•} × Y n−2 . Each element in this family encodes a reduction T ∼ Tij : there is one map from Y n−2 corresponding to T and another map from Y n−2 corresponding to Tij . For the second map the diagonal y 7→ (y, y) is used to define the new labelling. So for each such tree T with → − edge ij there are a pair of maps // NAPY (n). (3.3.7) Y n−2 In identifying the two images of each point we are taking the coequaliser of this → − diagram. But we have such an identification for each tree T with an edge ij where i is not the root. So we have a diagram with a copy of Y n−2 for each such pair → − (T, ij ) and two arrows from each copy to a single copy of NAPY (n). The colimit of this diagram is the space given by making all identifications T ∼ Tij – that is, the colimit is BCactY (n). We will use G to denote the group π1 (Y, P ). The fundamental groupoid functor π1 respects colimits and products, so in particular respects polynomial functors meaning that π1 (NAPY , NAPP ) ∼ = NAPG . Furthermore (3.3.8)

BCactG (n) := π1 (BCactY (n), BCactP (n))

is given by the colimit of the diagram which consists of a single copy of NAPG (n) → − and a copy of Gn−2 for each pair (T, ij ). It now remains to compute this colimit. Restricting ourselves to trees with root 1, we have that BCactY (n)1 is the colimit of the diagram where NAPY (n) is replaced by NAPY (n)1 and we only → − include pairs (T, ij ) where the root of T is 1. Since BCact• (n)1 is a single point, BCactY (n)1 is connected and BCactG (n)1 has a single object and so may be viewed as a group. We will now examine the effect of coequalisers on morphisms. A generic morphism of NAPG (n)1 consists of a rooted tree T ∈ RT(n)1 with edge labels ge ∈ G for each e ∈ E(T ). But since such elements belong to a component of NAPG (n) isomorphic to Gn−1 they can be rewritten as the product of n − 1 elements, one for each edge. The element corresponding to e ∈ E(T ) is given by labelling edge e by ge and every other edge of T by the identity. We will denote such an element by → − − , which is the tree T with edge ij labelled by g ∈ G. g(T,→ ij ) → ∈ NAPG (n)1 be The coequalisers encode reductions just as before. Let g(T,− vw) → − − → a generator where g ∈ G, T ∈ RT(n) and vw ∈ E(T ) is any edge. Let ij be 1

another edge, this time we ask that i is not the root 1; this will be the edge we will → − reduce over. As before let ki be the unique incoming edge to i and let Tij be the − → → − → may be reduced when tree given by cutting ij and adding kj. The element g(T,− vw) − → − → − − → → = → the label of ij is the identity, that is if vw 6= ij : in the case that − vw ki the


12

→ − − → reduced tree has edges ki and kj labelled by g and the remaining edges labelled by → − the identity. In all other cases the single edge ki is labelled by g with the remaining → − → = ki we have g → − =g → − .g − → and edges labelled by the identity. So if − vw (T, ki) (Tij , ki) (Tij ,kj) → − − =g → − . Remember that ij must be a reducible edge. otherwise g(T,→ ki) (Tij , ki) The reductions above allow (using the fact that reduction reduces the level) any − where T is element in BCactG (n) to be written as a product of elements g(T,→ ij ) a tree with no identity labelled reducible edges. The possibilities are that T is a corolla and so i = 1, or that T is the tree with n − 2 edges emanating from the root → − 1 and the only other edge being ij . So for each pair (i, j) where i, j ∈ [n], i 6= j and → − j 6= 1, there is a unique tree T such that the pair (T, ij ) is irreducible. Therefore g − by α we may denote the element g(T,→ ij and these elements generate the group ij ) BCactG (n)1 . → − − Let T ∈ RT(n), ij be any edge and g ∈ G, we may write the element g(T,→ ij ) as a monomial in the generators above as follows. Let Aij be the set of vertices v → − which may be joined by a directed path from i to v starting in the edge ij . Then − reduces to the product g(T,→ ij ) Y g (3.3.9) αiv . v∈Aij

It remains to find the relations between the generators. Some of the relations are contributed by the components of NAPG (n)1 corresponding to the irreducible pairs → − g gh h (T, ij ). The relations αij .αij = αij come from their respective components, these g h account for (3.3.3). Then there are the relations [α1i , α1j ] for i 6= j, which exist in the component of NAPG (n)1 corresponding to the corolla, these account for some of the relations in (3.3.4), specifically the relations for i = 1. Denote by T (ij) the − → → − tree with edges 1k for k 6= j and edge ij , then this contributes the relations h i g → (3.3.10) αij , h(T (ij),− = e. 1k) h But the second element is reducible: in the case k 6= i it reduces to α1k , whilst in h h the case k = i it reduces to α1i .α1j . However these are not all of the relations, additional commutation relations come → − → − from other trees T . Let T (ij, ik) be the tree with the edges ij and ik and edges → − 1l for l 6= j, k. This tree encodes commutator brackets h i − ,h → − (3.3.11) g(T (ij,ik),→ = e, ij ) (T (ij,ik), ik) g h the elements reduce to αij and αik respectively. Similarly for distinct i, j, k, l 6= 1 → − → − −→ let T (ij, kl) be the tree with edges ij and kl and edges 1m for m 6= j, l; as above this encodes a commutator relation: h i g h − ,h → − (3.3.12) g(T (ij,kl),→ = αij , αkl = e. ij ) (T (ij,kl), kl)

− → → − → − Finally let T (ij, jk) be the tree with edges ij and jk and edges 1l for l 6= j, k. This tree gives the commutator relations h i − ,h − → (3.3.13) g(T (ij,jk),→ = e. ij ) (T (ij,jk),jk)


13

g g h .αik . This The second element reduces to αjk and the first to the product αij accounts for all of the relations in the statement of the proposition. To show that the stated relations are sufficient to present the group we need to show that the commutator relations h i − ,h → − (3.3.14) g(T,→ =e ij ) (T, kl)

− → − → hold for each tree T and each pair of edges ij , kl. Let Aij and Akl be the sets of vertices which index the respective decompositions of the form (3.3.9). If Aij and Akl are disjoint then commutator relations of the form (3.3.4) and (3.3.5) show that all the constituent irreducible elements commute with one another. In the case that Aij and Akl do intersect there must be either a directed path from i to k or from k to i. Assuming the former we find that Aij contains Akl . We now show that each h − . Since both k and v are in Aij the element αkv for v ∈ Akl commutes with g(T,→ ij ) g g − , the relation (3.3.6) means that αh commutes with this αik .αiv is a term in g(T,→ kv ij ) g term. The remaining terms are of the form αiw for w 6= k, v which also commutes h h → − − ; and therefore h with αkv . Therefore αkv commutes with the element g(T,→ ij ) (T, kl) commutes with it as well. Therefore the relations (3.3.3)-(3.3.6) suffice to present BCactG (n)1 .

We have already seen that π1 (BCactY , BCactP ) is an operad, to give the composition maps we need only describe the compositions on the generating morphisms. In fact since we have g ◦i h = (g ◦i e).(e ◦i h) we need only describe the compositions of generators with identity maps. Proposition 3.4. Let (Y, •) be a path connected pointed space and let G be its fundamental group. The operad structure on π1 (BCactY , BCactP ) is given on g generating morphisms as follows: let αij ∈ BCactG (n)r and e ∈ BCactG (m)s be the identity morphism. For a ∈ [m] define i0 = i + a − 1 and j 0 = j + a − 1, then we have (3.3.15)

g e ◦a αij = αig0 j 0 .

For b ∈ [n] define i00 to be i if i < b, to be i + m − 1 if i > b and i + s − 1 if i = b; define j 00 similarly. For each l ∈ [m] define l00 to be l + b − 1. Then we have (Q m g if b = j, and g l=1 αi00 l00 (3.3.16) αij ◦b e = αig00 j 00 otherwise. − → → − Proof. Let T (ij)r be the tree with root r, the edge ij and (n − 2) edges rk (if i = r → − g then this is a corolla). Then αij is represented by the tree T (ij)r with ij labelled − → by g ∈ G. Let Cs be the corolla with root s and m − 1 edges sk. When all of the edges are labelled by the identity e ∈ G then this represents the identity e of BCactG (m)s . g To compute e ◦a αij we compose trees to get Cs ◦a T (ij)r and then reduce using → − Equation (3.3.9). The unique labelled edge ij of T (ij)r is a leaf and hence it is −→ also a leaf of Cs ◦a T (ij)r , although now the edge is i0 j 0 . Since it is a leaf it reduces g to αi0 j 0 as required. To compute αg ◦b e is a little more complicated as it depends on the value of b. → − If b 6= j then the leaf ij is still a leaf of T (ij)r ◦b Cs and so the same argument


14

applies to give the reduction to αig00 j 00 . However if b = j then the tree consists −−→ of the edge i00 j 00 , another n − 2 edges emanating from the root and m − 1 edges −− → −−→ j 00 l00 . The only labelled edge is i00 j 00 and the set Aij of vertices ‘above i’ consists → − of the vertex j 00 = s00 and the vertices l00 for each edge jl ∈ Cs . An application of Equation (3.3.9) serves to finish the proof. Remark 3.6. The groups BCactG (n)r act faithfully on the free product G∗n . We will write this free product as G1 ∗ . . . ∗ Gn where each group is isomorphic to G in g order to distinguish between different factors. The element αij acts on the factors as follows ( −1 hgi if h ∈ Gj and where gi = g in Gi and g (3.3.17) αij (h) = h if h ∈ Gk for k 6= j. In [18] the closely related spaces of unbased cacti CactY were studied and it was shown that when Yi is a classifying space for Gi then CactY is itself a classifying space for a certain group of automorphisms. As a consequence of Theorems 4.3 and 6.9 we see that (3.3.18) H∗ (BCactY ) ∼ , = Perm ◦NAP H(Y )

whereas in [18] it is shown that (3.3.19)

H∗ (CactY ) ∼ = Com ◦NAPH(Y ) .

This last isomorphism could also be shown using the methods of reduction used in this paper, although CactY is not an operad. 3.4. Relationships with other topological operads. The pure braid group on n strands, Pn is known to be a subgroup of the group P Σn ∼ = π1 (CactY (n)) of partial conjugations of the free group on n letters. This inclusion may be realised by a construction involving cacti. In [20] various (quasi-)operads of cacti are discussed; these are different to the operad BCactS 1 in that the cacti are planar and unbased. We will take PlCact to be the spineless and normalised varieties of cacti from [20]. This quasi-operad is quasi-isomorphic to the little discs operad and so in particular the fundamental group π1 (PlCact(n)) is the pure braid group Pn . There is an Sn -equivariant map (3.4.1)

PlCact(n) → CactS 1 (n)

defined by the map which forgets the planar structure of a planar cactus leaving a cactus product of circles as defined in (3.2.1); on fundamental groups this gives the inclusion Pn → P Σn . The operad compositions of BCactS 1 and PlCact are not closely related, this may be seen by examining the homology operads which are BCactH∗ (S 1 ) as defined in the next section and the Gerstenhaber operad e2 . However both families of cacti are related by a third operad which ‘contains’ both. Let LR(n) be the space of smooth, disjoint embeddings of n copies of the filled in torus, or ring R = S 1 × D2 into itself — this is naturally an operad. The little discs operad consists of disjoint embeddings of copies of a disc D2 into itself and can be mapped into the little rings operad LR by applying idS 1 × (−) to the embeddings. The image of the little discs operad involves little rings which wind around the large ring once. Meanwhile the operad BCactS 1 is related to the connected components of embeddings in which one little ring, the root winds


15

around the large ring once; the remaining rings do not wind around the large ring and all of the rings are unknotted and unlinked. The fundamental groups of these connected components contain π1 (BCactS 1 ) ∼ = BCactZ as a suboperad. There are additional elements not in the suboperad given by little rings circling through the large ring along with smooth endomorphisms of R. 4. The homology operads So far we have described operads NAPY and BCactY in the “geometric” setting. Both families also have versions existing in the “linear” setting, so for any graded vector space D there exists an operad NAPD , whereas in the case of the based cacti there is a subtlety, we require a graded augmented cocommutative coalgebra C to define BCactC . The “geometric” and “linear” versions are closely related via the homology functor which sends a topological space to its homology groups with coefficients in the base field k. In this section, we shall describe these operads via constructions with decorated rooted trees, and later in section 6, we shall descibe them via generators and relations, and show that it in fact each of them has a quadratic Gr¨ obner basis of relations. 4.1. The linear operad NAPD . Let D be a graded vector space (over some field k). Recall that in (3.1.1) we described NAPY (n) as disjoint union of direct products of copies of Y . Then in Remark 3.1 we gave a polynomial diagram (3.1.10) realising NAPY (n) as a polynomial functor in Y . Let D be a graded vector space and define NAPD via the same polynomial diagram in the category of graded vector spaces: M (4.1.1) NAPD (n) = D⊗(n−1) . T ∈RT(n)

Equivalently NAPD (n) is the vector space spanned by rooted trees with vertex set [n] and edge labels in D, subject to linearity in each edge label. The set based description of the NAPY operad works on the level of polynomial functors and so suffices to show that NAPD is an operad. However great care must be taken to keep track of the signs induced by the symmetry σ from the symmetric monoidal category (gVect, ⊗, σ, k) of graded vector spaces. In order to do this we must assign for each term D⊗n−1 in the sum (4.1.1) a reference ordering of the factors. This requires assigning to each tree T ∈ RT(n) a total ordering on the set of edges E(T ). Let T be such a tree and let i be its root. Since each vertex has a unique incoming edge except for the root which has none, the set of edges E(T ) is in bijection with the set of non-root vectices [n] − i. We take the ordering of E(T ) from the natural ordering of [n] − i. So for instance the pair   2 ^== 3 2 ^== @3 = = @   y z  (4.1.2) , x ⊗ y ⊗ z . 1O 1O   represents the Y -tree x

4

4

The order of x, y and z in the tensor product is determined by the order of the edges. The first step in giving the operad structure is to describe the action of the symmetric group Sn on NAPD (n). For instance applying the permutation (24) to


16

the Y -tree considered in (4.1.2) we get 

(4.1.3)

 2 ^== @3 =   (24).  , x ⊗ y ⊗ z 1O  = 4  3 ^== 4 3 ^== @4 = @ =   y |y||z| z , (23)x ⊗ y ⊗ z  . = 1O 1O  = (−1)  

x

2

2

The signs involved in the composition T ◦i T 0 for T ∈ NAPD (n) and T 0 ∈ NAPD (m) are more easily accounted for. This is because the edges within the righthand tree T 0 are not reordered within T ◦i T 0 and so the sign depends on the total degree |T 0 | and not on the individual edges. The edges of T 0 are ‘moved past’ − → − → the edges jk ∈ E(T ) for which k > i. Hence if yjk is the labelling of jk the sign change is given by X |T 0 | |yjk | (4.1.4)

(−1)

− → jk∈E(T )|k>i

.

Proposition 4.1. The homology operad H∗ (NAPY ) with coefficients in the base field k is isomorphic to the linear operad NAPH∗ (Y ) . Proof. With field coefficients the homology functor H∗ from topological spaces to graded vector spaces respects products and coproducts and so is compatible with polynomial functors. The explicit expression of this is a M H∗ (Y )⊗E(T ) . (4.1.5) H∗ (NAPY (n)) ∼ Y E(T ) ∼ = = H∗ T ∈RT(n)

T ∈RT(n)

4.2. The linear operads of based cacti. Let C be an augmented cocommutative coalgebra and write its splitting as k 1 ⊕C. The operad BCactC will be a quotient of the operad NAPC , this is a parallel of the set-based versions. Let T ∈ NAPC → − be a C-labelled rooted tree and suppose that it has an edge ij with the label 1 → − and suppose further that i is not the root of T , as before we will call the edge ij → − reducible. Let k be the unique vertex such that ki is an edge and let c be the label → − of ki. We define T 0 to be the unlabelled rooted tree created by removing the edge − → → − ij and replacing it by kj and denote by T 0 (a, b) the edge labelled rooted tree based → − on T 0 where the edge labels are inherited from those of T except for ki which is − → labelled by a and kj which is labelled by b. Finally we define Tij to be the sum X (4.2.1) (−1)|c(2) |g T 0 (c(1) , c(2) ), where g is the sum of degrees (4.2.2)

X − → xy|i